let Z be open Subset of REAL ; :: thesis: ( Z c= dom (exp_R (#) sin ) implies ( exp_R (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) sin ) `| Z) . x = (exp_R . x) * ((sin . x) + (cos . x)) ) ) )
A1: ( sin is_differentiable_on Z & exp_R is_differentiable_on Z ) by FDIFF_1:34, SIN_COS:73, TAYLOR_1:16;
assume A2: Z c= dom (exp_R (#) sin ) ; :: thesis: ( exp_R (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) sin ) `| Z) . x = (exp_R . x) * ((sin . x) + (cos . x)) ) )

now

let x be Real; :: thesis: ( x in Z implies ((exp_R (#) sin ) `| Z) . x = (exp_R . x) * ((sin . x) + (cos . x)) )
assume x in Z ; :: thesis: ((exp_R (#) sin ) `| Z) . x = (exp_R . x) * ((sin . x) + (cos . x))
hence ((exp_R (#) sin ) `| Z) . x = ((sin . x) * (diff exp_R ,x)) + ((exp_R . x) * (diff sin ,x)) by A2, A1, FDIFF_1:29
.= ((sin . x) * (exp_R . x)) + ((exp_R . x) * (diff sin ,x)) by TAYLOR_1:16
.= ((sin . x) * (exp_R . x)) + ((exp_R . x) * (cos . x)) by SIN_COS:69
.= (exp_R . x) * ((sin . x) + (cos . x)) ;
:: thesis: verum

end;

hence ( exp_R (#) sin is_differentiable_on Z & ( for x being Real st x in Z holds
((exp_R (#) sin ) `| Z) . x = (exp_R . x) * ((sin . x) + (cos . x)) ) ) by A2, A1, FDIFF_1:29; :: thesis: verum